Fritz Zwicky was born in Bulgaria in 1898 to Swiss parents. He worked at the California Institute of Technology for most of his life. Zwicky is responsible for several important discoveries in the field of astrophysics, but his most notable discovery was that of dark matter.
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| Fritz Zwicky Image Credit: http://www.cosmotography.com/images/starburst/images/fritz_zwicky.jpg |
In 1933, Zwicky, made an important observation that gave the first implications for the existence of dark matter. While observing the Coma Cluster (a cluster of gravitationally bound galaxies) and applying Virial Theorem to the galaxies in the cluster, Zwicky found that the velocities of the galaxies were much larger than expected from the expected mass in the cluster, implying that there must be more mass (or gravitational potential) in the cluster that was unseen. It is due to the unobservable nature of this mass that we call it dark matter.
Fortunately, we have enough of an understanding of the Virial Theorem to follow the calculations Zwicky made to reach his conclusions.
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| Coma Cluster Image Credit: http://upload.wikimedia.org/wikipedia/commons/7/7d/Ssc2007-10a1.jpg |
First, Zwicky made the reasonable assumption the the cluster was in a stationary state (neither expanding nor contracting), meaning that it was virialized. As we have gone over before, this means that the total kinetic energy is equal to half of the total potential energy in the system.
\(K=-\frac{1}{2}U\)
He also assumed that the density, \(\rho\), was constant, which while not true, is okay since he was only trying to make an estimation. The radius, R, of the Coma Cluster is approximately one million light years, which is equal to \(10^{24}\) cm. The mass, M, was also estimated by finding the number of galaxies and multiplying by the mass of a galaxy:
\(M=800\times 10^9M_\odot\times 2\times 10^{33}\text{ g M}_\odot^{-1}\approx 1.6\times 10^{45}\) g
So the potential energy, assuming spherical shape, is given by:
\(U=-\dfrac{3GM^2}{5R}\)
\(U=-\dfrac{3\times (6.674\times 10^{-8}\text{ cm}^3\text{ g}^{-1}\text{ s}^{-2})\times (1.6\times 10^{45}\text{ g})^2}{5\times 10^{24}\text{ cm}}\)
\(U\approx -10^{59}\text{ g cm}^2\text{ s}^{-2}\)
Where G is the gravitational constant.
Since we know that the Coma Cluster is virialized, we can use Virial Theorem to calculate the average velocity:
\(K=-\frac{1}{2}U\)
\(\frac{1}{2}M\langle v\rangle ^2=-\frac{1}{2} (-10^{59})\)
\(\langle v\rangle =\sqrt{\dfrac{10^{59}\text{g cm}^2\text{ s}^{-2}}{1.6\times 10^{45}\text{ g}}}\)
\(\langle v\rangle \approx 8\times 10^6\text{ cm s}^{-1}=80\text{ km s}^{-1}\)
However, through observations on the Doppler shifts in the spectrum of the galaxies of the Coma Cluster, Zwicky found that there was a large spread of velocities in the cluster, with average velocities of at least 1000 km/s, and some velocities even as large as 1500 to 2000 km/s. In order for Virial Theorem to come up with velocities this large, we would need to have a total mass at least 400 times larger than our previous estimate. This is a very large difference, however, so it seems unlikely that our estimate based on observations of galaxies could be this far off.
There are four possible explanations discussed by Zwicky as to why we might be observing velocities that are much larger than expected. His first, and most favorable, explanation is the existence of dark matter. There could be some form of matter in the universe that is unobservable but has gravitational effects. While today we have a large amount of evidence that speaks to the validity of this claim, at the time, Zwicky's idea of dark matter was not accepted, and would not be accepted for another fifty years.
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| Coma Cluster Image Credit: http://upload.wikimedia.org/wikipedia/commons/0/02/Coma_Cluster_of_Galaxies_%28visible%2C_wide_field%29.jpg |
Zwicky's second explanation for this discrepancy in theory and observation was that it was possible that the Coma Cluster was not virialized, and that instead, the total kinetic energy was equal to the total potential energy.
\(K=-U\)
But this only accounts for a factor of two from our previous findings. So there still remains the problem of a large proportion of mass that is unaccounted for.
Another possibility is that the observations were correct, and that Zwicky's estimate for mass based on the luminous (visible) matter was valid. If this were true, however, the large velocities of the constituent galaxies in the cluster would cause them to break away from each other's gravitational pull, and the cluster would fall apart, leaving the 800 galaxies free, flying through space at their velocities of 1000 to 2000 km/s. If this were the case, however, we would probably be able to observe other free galaxies that are not associated with clusters (called field galaxies) which have similar velocities; but thus far, none have been seen to even have velocities over 200 km/s. So it seems unlikely that this possibility would hold true.
His last explanation was that perhaps the observations of the velocities were incorrect. It is possible that the change in observed wavelength of light was not caused by motion (it was not a Doppler Shift), but instead by a gravitational force that was distorting the light observed. This kind of shift in wavelength is called gravitational redshift, or Einstein redshift. The equation to find redshift, z, is given as:
\(z=\dfrac{\Delta \lambda}{\lambda}\approx -\dfrac{GM}{c^2R}\)
Where c is the speed of light, \(\lambda\) is the emitted wavelength of the light, and \(\Delta\lambda\) is the shift in wavelength (the difference of the observed wavelength and emitted wavelength). From this equation, using our previous estimate for mass, we can solve for the redshift, and then using the relation, \(v\approx cz\), we can solve for the velocities of the galaxies in the Coma Cluster.
\(z\approx \dfrac{(6.674\times 10^{-8}\text{ cm}^3\text{ g}^{-1}\text{ s}^{-2})\times (1.6\times 10^{45}\text{ g})}{(3\times 10^{10}\text{ cm s}^{-1})^2\times 10^{24}\text{ cm}}\)
\(z=3.5\times 10^{-8}\)
Note that Zwicky's calculation may have been more careful, so I reported his calculation for z.
Now solving for velocity, v:
\(v=cz\approx (3\times 10^{10}\text{ cm s}^{-1})(3.5 \times 10^{-8})\approx 1000\text{ cm s}^{-1}=10\text{ m s}^{-1}\)
This certainly doesn't explain the large velocities observed. We could, instead, work the other way, starting with the large velocities and solving for the mass necessary to create this gravitational redshift. However, this method would require a mass that is much greater than before, which again implies that there must be a large proportion of dark matter.
From Zwicky's discussion of the possible explanations for the large velocities of galaxies in the Coma Cluster, it is apparent that there must be some form of matter that is unobservable but has gravitational effects which could cause these larger velocities. The nature of this dark matter is still unknown, but we have since found evidence that suggests that dark matter is the predominant form of matter in our universe.
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This is a wonderful review of the Zwicky paper!
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